d3RPG – Exploding Dice and Probabilities

I recently wrote to introduce d3RPG as a concept. I’ve been working on it a lot more than any other projects recently, because several others (that I haven’t introduced at all) are lacking in playtesting. There’s only so many opportunities to force my regular gaming group into testing things for me, but there’s always a chance for me to write more when there are no minions – uh, friends – available.

So today, we’re going to take a look at something called Exploding Dice, how they affect Probability, and how this works in d3RPG.

Statistics…? While still trying to build a core audience? Ugh.

Dice Notation

At the end of my last post, I promised more information about Exploding Dice, so that’s today’s topic. The concept of Exploding Dice is fairly straight forward, though, so I’ll also cover how this idea affects the probabilities encountered in d3RPG.

Firstly, though, we need to understand dice notation. Those of us familiar with RPGs are usually pretty comfortable using all these weird shaped pieces of plastic. But to many people, a six-sided die is the only die they’ve every used. News Flash: Dice can have more (or even less) than six sides! So how do we easily tell people what they need to roll?

“Roll your damage. It’s three six-sided dice, plus your strength modifier of 4” is not an overly-complicated sentence to parse or to speak. But in the written format, it takes an unnecessary amount of time and space to write. Instead, we use notation. The same instruction could be notated as “Damage: 3d6+4”. Much cleaner and easier. This can be abstracted to “XdY+Z”, where:

X = the number of dice being rolled. Occasionally if you are rolling a single die, the “1” is omitted, but it is good practice to include it to remove doubt or confusion.

Y = the number of sides on the dice. A common – but by no means exhaustive – list includes d4, d6, d8, d10, d12 and d20.

Z = any other modifiers that apply. Depending on the system you are playing in, these may be based on many things including your weapon being used, your strength or dexterity score or the enchantments in place on your weapon.

Now, whenever I use dice notation, you’ll hopefully have no issues understanding what all that gobbledygook means.

What are Exploding Dice?

I’m sure the idea exists in many other game systems, but I must recognise Savage Worlds as the ruleset that introduced me to the idea. If you are playing with Exploding Dice, then whenever you roll a die and it comes up at it’s maximum value (6 on a d6, for example), you get to reroll that die, and add the results together. That’s it, really. Super simple, but really fun in game.

Let’s say you’re rolling 2d6, and you roll a 4 and a 6. That’s a total of 10 – well above average for 2d6 (which is 7, by the way). But because the 6 you rolled is the maximum possible on that die, you get to roll it again. This reroll might give you a 3. You now have 4 + 6 (+3) = 13 total, on just two dice. You can easily and quickly see (probably even without the benefit of my example) that this total of 13 is higher than would be possible if you were just rolling dice as per normal. 2d6 can only roll up to 12, after all.

And THAT is where the excitement of Exploding Dice comes in. In a system like Savage Worlds, there is a possibility – no matter how small – that you could succeed on ANY roll, because there is no limit to the number of times a die can explode!

Sticking with the familiar d6’s, one theoretical roll with a single Exploding Die could be 6 (+6) (+6) (+3) = 21. A single six-sided die giving a result of 21?! Impossible! Not with Exploding Dice, though, just highly improbable (Rolling three 6’s in a row is a 1:216 chance, or a little less than 0.5%). It can be crazy fun!

To notate Exploding Dice, we just add an exclamation point to the dice formula: “2d8!” means to roll two eight-sided dice, rerolling and adding the results of any 8’s you roll.

Exploding Dice in d3RPG

When I decided I wanted to use d3’s for this game (it’s in the name, after all) it was pretty obvious to me pretty quickly that if players were to roll any specific quantity of d3’s, then the range of results was going to be fairly narrow. There could only possibly be a three-fold difference between the minimum roll, and the maximum.

3d3 Min = 3, Max = 9

6d3 Min = 6, Max = 18

10d3 Min = 10, Max = 30

etc

This didn’t strike me as particularly exciting. But at the same time, I didn’t want a huuuge range as is possible in most d20-based systems.

When I considered the use of Exploding Dice, it suddenly gave me a wider range of possible outcomes. Initially, and until now, I have been considering limiting the “explode depth” to 1 (That is, each die can only explode once). During further number crunching while writing this very blog post, though, I’m starting to think that might be an artificial limit. It bears thinking about further.

Instead of trying to describe the difference between 3d3 and 3d3! in words, a pretty little chart might do a better job. A picture is worth a thousand words, after all.

Chart showing probability of rolling a given value – incorporates both regular and exploding dice. Click for larger image.

You can see that regular dice rolls (orange) cluster in a neat little bell curve, while the Exploding Dice (blue) have a much longer tail, showing how higher values can be obtained at the expense of making individual results far less likely.

Of course, most RPGs require you to roll against a target of some description – the most basic example is when making an attack against a creature. Whenever you roll to attack, you are usually aiming to beat their Armor Class, Toughness, or other similar term, depending on the system you are playing. When I combined a range of potential bonuses (bonii?) to compare them to various possible results, this is what I got:

You can see in the chart above that I have highlighted the “Target 10” row. After considering the probabilities in conjunction with the “difficulty” I wanted in the game I decided a good default target would be 10. Not only does it make sense from the probabilities perspective, but it’s an easy number to remember. Of course, the Storyteller (aka, Gamemaster, Judge etc… take your pick) will modify the target regularly, and various features of targeted foes (Skills and Perks – more on those in another post) will adjust the target too. If you were only ever rolling against a Target of 10, things would get pretty stale pretty quickly.

When interpreting the chart, simply find the column that shows the total Bonuses and Penalties that apply to the roll, and the row that shows the Target. Where they intersect will tell you the character’s chance of succeeding in their attempts at… well, doing whatever they’re trying to do.

Phew!

And that’s really it for now. I don’t know how enlightening this will be to those of you who haven’t seen anything else in the core rules yet (That is, the population of the world, minus myself and one other person). But I felt an outline of the central mechanic was probably warranted early on in the piece.

Before I actually go, though, a quick shoutout. Manually calculating all the probabilities for Exploding Dice would give me a case of Exploding Brain. Luckily, there’s an awesome website called anydice.com which does ALL the maths for you. I literally could not have gotten as far along with designing this game as I have without them. You should totally check them out.

Next time I write about d3RPG I think we should take a look at another core aspect of (pretty much) any RPG: Attributes and Skills. Come back and join the fun then!